The Uniqueness of a Correction to Interaction Parameter Formalism in a Thermodynamically Consistent Manner

Polynomial representation of partial excess Gibbs energy (i.e., activity coefficient) in multicomponent dilute solution has been widely used after Wagner. Although Wagner’s Interaction Parameter Formalism has been known to be only strictly valid at infinite dilute concentration, it has been used often, even at finite concentration, due to its mathematical simplicity. Nevertheless, several attempts have been made to correct the formalism to be thermodynamically consistent at finite concentration. Among those, Unified Interaction Parameter Formalism proposed by Pelton and Bale, which may be considered as an extension of Darken’s quadratic formalism, has obtained much attention. However, there have been much confusion and debate about the way of the correction. Recently, a thermodynamic analysis was reported that there are infinite numbers of ways to correct Wagner’s formalism to be thermodynamically consistent, which may prevent one from using the Unified Interaction Parameter Formalism with confidence. In the present article, the correction to the Wagner’s formalism is discussed by revisiting Darken’s condition of the thermodynamic consistency. It is shown that the correction to the Wagner’s formalism can be made uniquely. It is pointed out that to ensure the thermodynamic consistency among the activity coefficients of all components, Gibbs–Duhem relation and Maxwell relation among all components including solvent–solute, must be obeyed. Derived expressions for activity coefficients of all components by path-independent integration are also shown to be the same as those obtained by differentiating a corresponding integral excess partial Gibbs energy.